TSTP Solution File: SET008^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SET008^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n102.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:29:54 EDT 2014
% Result : Timeout 300.09s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SET008^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n102.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 07:29:11 CDT 2014
% % CPUTime : 300.09
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x228bf80>, <kernel.Type object at 0x226d8c0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (X:(a->Prop)) (Y:(a->Prop)), (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) (fun (Xx:a)=> False))) of role conjecture named cBOOL_PROP_78_pme
% Conjecture to prove = (forall (X:(a->Prop)) (Y:(a->Prop)), (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) (fun (Xx:a)=> False))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (X:(a->Prop)) (Y:(a->Prop)), (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) (fun (Xx:a)=> False)))']
% Parameter a:Type.
% Trying to prove (forall (X:(a->Prop)) (Y:(a->Prop)), (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) (fun (Xx:a)=> False)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) (fun (x:a)=> ((and ((and (X x)) ((Y x)->False))) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found x01:(P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P0 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found x01:(P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P0 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found x01:(P False)
% Found (fun (x01:(P False))=> x01) as proof of (P False)
% Found (fun (x01:(P False))=> x01) as proof of (P0 False)
% Found x01:(P False)
% Found (fun (x01:(P False))=> x01) as proof of (P False)
% Found (fun (x01:(P False))=> x01) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found x01:(P False)
% Found (fun (x01:(P False))=> x01) as proof of (P False)
% Found (fun (x01:(P False))=> x01) as proof of (P0 False)
% Found x01:(P False)
% Found (fun (x01:(P False))=> x01) as proof of (P False)
% Found (fun (x01:(P False))=> x01) as proof of (P0 False)
% Found x:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))):(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) False)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) False)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) False)
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) False)
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found x:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) False)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) False)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) False)
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) False)
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found x:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))):(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) (fun (x:a)=> ((and ((and (X x)) ((Y x)->False))) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) False)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) False)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) False)
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) False)
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found x:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) False)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) False)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) False)
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) False)
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x:(P0 b)
% Instantiate: b:=(fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))):(a->Prop)
% Found (fun (x:(P0 b))=> x) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (fun (P0:((a->Prop)->Prop)) (x:(P0 b))=> x) as proof of ((P0 b)->(P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))
% Found (fun (P0:((a->Prop)->Prop)) (x:(P0 b))=> x) as proof of (P b)
% Found x3:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (fun (x3:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))=> x3) as proof of (P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (fun (x3:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))=> x3) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) (fun (x:a)=> ((and ((and (X x)) ((Y x)->False))) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x:(P0 b)
% Instantiate: b:=(fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))):(a->Prop)
% Found (fun (x:(P0 b))=> x) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (fun (P0:((a->Prop)->Prop)) (x:(P0 b))=> x) as proof of ((P0 b)->(P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))
% Found (fun (P0:((a->Prop)->Prop)) (x:(P0 b))=> x) as proof of (P b)
% Found x3:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (fun (x3:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))=> x3) as proof of (P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (fun (x3:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))=> x3) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found x0:(P ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Instantiate: b:=((and ((and (X x)) ((Y x)->False))) (Y x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x0:(P ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Instantiate: b:=((and ((and (X x)) ((Y x)->False))) (Y x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found x0:(P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Instantiate: b:=((and ((and (X x)) (not (Y x)))) (Y x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x0:(P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Instantiate: b:=((and ((and (X x)) (not (Y x)))) (Y x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found x2:(P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P2 (fun (Xx:a)=> False))
% Found x2:(P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P2 (fun (Xx:a)=> False))
% Found x2:(P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P2 (fun (Xx:a)=> False))
% Found x2:(P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P2 (fun (Xx:a)=> False))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x3:(P (fun (Xx:a)=> False))
% Found (fun (x3:(P (fun (Xx:a)=> False)))=> x3) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x3:(P (fun (Xx:a)=> False)))=> x3) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (not (Y x)))) (Y x))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (not (Y x)))) (Y x))))
% Found x2:(P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P2 (fun (Xx:a)=> False))
% Found x2:(P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P2 (fun (Xx:a)=> False))
% Found x2:(P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P2 (fun (Xx:a)=> False))
% Found x2:(P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x2:(P1 (fun (Xx:a)=> False)))=> x2) as proof of (P2 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x3:(P (fun (Xx:a)=> False))
% Found (fun (x3:(P (fun (Xx:a)=> False)))=> x3) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x3:(P (fun (Xx:a)=> False)))=> x3) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found x01:(P1 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P1 ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P1 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P1 ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P2 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found x01:(P1 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P1 ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P1 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P1 ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P2 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found x01:(P1 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P1 ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P1 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P1 ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P2 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found x01:(P1 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P1 ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P1 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P1 ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P2 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found x0:(P False)
% Instantiate: b:=False:Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found x0:(P False)
% Instantiate: b:=False:Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) (fun (x:a)=> ((and ((and (X x)) ((Y x)->False))) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x0:(P False)
% Instantiate: b:=False:Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found x0:(P False)
% Instantiate: b:=False:Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found x02:(P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x02:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x02) as proof of (P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x02:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x02) as proof of (P0 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x02:(P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x02:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x02) as proof of (P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x02:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x02) as proof of (P0 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x0:(P0 b)
% Instantiate: b:=((and ((and (X x)) (not (Y x)))) (Y x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 ((and ((and (X x)) (not (Y x)))) (Y x))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x0:(P0 b)
% Instantiate: b:=((and ((and (X x)) (not (Y x)))) (Y x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 ((and ((and (X x)) (not (Y x)))) (Y x))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found x0:(P False)
% Instantiate: b:=False:Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found x0:(P False)
% Instantiate: b:=False:Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) (fun (x:a)=> ((and ((and (X x)) (not (Y x)))) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:(P False)
% Instantiate: b:=False:Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found x0:(P False)
% Instantiate: b:=False:Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x02:(P False)
% Found (fun (x02:(P False))=> x02) as proof of (P False)
% Found (fun (x02:(P False))=> x02) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x02:(P False)
% Found (fun (x02:(P False))=> x02) as proof of (P False)
% Found (fun (x02:(P False))=> x02) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x02:(P False)
% Found (fun (x02:(P False))=> x02) as proof of (P False)
% Found (fun (x02:(P False))=> x02) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found x02:(P False)
% Found (fun (x02:(P False))=> x02) as proof of (P False)
% Found (fun (x02:(P False))=> x02) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found x0:(P0 b)
% Instantiate: b:=False:Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 False)
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 False))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found x0:(P0 b)
% Instantiate: b:=False:Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 False)
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 False))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found x0:(P ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Instantiate: b:=((and ((and (X x)) ((Y x)->False))) (Y x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found x0:(P ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Instantiate: b:=((and ((and (X x)) ((Y x)->False))) (Y x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x01:(P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P1 False)
% Found (fun (x01:(P1 False))=> x01) as proof of (P2 False)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found x02:(P False)
% Found (fun (x02:(P False))=> x02) as proof of (P False)
% Found (fun (x02:(P False))=> x02) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x02:(P False)
% Found (fun (x02:(P False))=> x02) as proof of (P False)
% Found (fun (x02:(P False))=> x02) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found x02:(P False)
% Found (fun (x02:(P False))=> x02) as proof of (P False)
% Found (fun (x02:(P False))=> x02) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found x02:(P False)
% Found (fun (x02:(P False))=> x02) as proof of (P False)
% Found (fun (x02:(P False))=> x02) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found x0:(P0 b)
% Instantiate: b:=False:Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 False)
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 False))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found x0:(P0 b)
% Instantiate: b:=False:Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 False)
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 False))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x0:(P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Instantiate: b:=((and ((and (X x)) (not (Y x)))) (Y x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x0:(P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Instantiate: b:=((and ((and (X x)) (not (Y x)))) (Y x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))):(((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) (not (Y x)))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x00:(P False)
% Found (fun (x00:(P False))=> x00) as proof of (P False)
% Found (fun (x00:(P False))=> x00) as proof of (P0 False)
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found x2:(P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(P (fun (Xx:a)=> False)))=> x2) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found x01:(P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P0 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found x01:(P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P0 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found x01:(P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P0 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found x01:(P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found (fun (x01:(P ((and ((and (X x)) (not (Y x)))) (Y x))))=> x01) as proof of (P0 ((and ((and (X x)) (not (Y x)))) (Y x)))
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))):(((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) ((and ((and (X x)) ((Y x)->False))) (Y x)))
% Found (eq_ref0 ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found ((eq_ref Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) as proof of (((eq Prop) ((and ((and (X x)) ((Y x)->False))) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found x01:(P False)
% Found (fun (x01:(P False))=> x01) as proof of (P False)
% Found (fun (x01:(P False))=> x01) as proof of (P0 False)
% Found x01:(P False)
% Found (fun (x01:(P False))=> x01) as proof of (P False)
% Found (fun (x01:(P False))=> x01) as proof of (P0 False)
% Found x01:(P False)
% Found (fun (x01:(P False))=> x01) as proof of (P False)
% Found (fun (x01:(P False))=> x01) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found x01:(P False)
% Found (fun (x01:(P False))=> x01) as proof of (P False)
% Found (fun (x01:(P False))=> x01) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) a
% EOF
%------------------------------------------------------------------------------